Vacuum Fluctuations Between Plates
Only standing-wave modes that fit exactly between the plates survive in the gap.
Force vs. Separation F(d)
Mode Spectrum & Energy Density
Standing-wave mode ladder between parallel plates
Two neutral conducting plates in vacuum attract each other due to restricted quantum vacuum fluctuations. Adjust separation, geometry, and material to explore the force.
Only standing-wave modes that fit exactly between the plates survive in the gap.
Standing-wave mode ladder between parallel plates
In 1948, Hendrik Casimir showed that two neutral, perfectly conducting parallel plates in vacuum experience an attractive force. Quantum electrodynamics predicts that even in perfect vacuum, electromagnetic fields undergo zero-point fluctuations — virtual photons popping in and out of existence. Between the plates, only modes with nodes at the surfaces are allowed (standing waves with wavelength λ_n = 2d/n). Outside, all modes exist. This asymmetry in vacuum energy density creates a net radiation pressure pushing the plates together. The result is F = −π²ℏcA/(240d⁴), a macroscopic force arising purely from quantum vacuum fluctuations — no charges, no classical fields needed.
The original Casimir formula assumes perfect conductors. Evgeny Lifshitz (1956) generalized it to real dielectric materials using the plasma model: the finite plasma frequency of a real metal means the plates become transparent at high frequencies, reducing the force from the ideal case. The correction factor η(δ, d) < 1 depends on the ratio δ/d where δ = c/ω_p is the plasma skin depth. For gold (δ ≈ 22 nm), the force is reduced by about 28% at d = 100 nm, and by about 50% at d = 50 nm. At very small gaps (~10 nm) the correction exceeds 90%. Temperature also modifies the force: at separations d ≫ ℏc/(k_B T) ≈ 7.6 μm at 300 K, the classical limit F = −ζ(3)k_B T A/(8πd³) applies.
The Casimir effect is crucial in nanotechnology and fundamental physics. In MEMS/NEMS devices, Casimir stiction causes moving parts to stick together at sub-micron gaps — a major reliability concern. Experiments by Lamoreaux (1997) and Mohideen (1998) confirmed the force to within 1% of theory. The effect is also central to dark energy cosmology (the cosmological constant problem), analog gravity models, and proposals for quantum levitation via geometry-engineered repulsive Casimir forces.
Drag the Plate Separation slider to change gap width. For parallel plates, the top panel shows the surviving standing-wave modes in the full gap. For sphere–plate and cylinder geometries, it switches to a local-gap view that highlights where mode suppression is strongest near closest approach. The bottom-left plot shows force vs. separation with the current operating point marked. The bottom-right plot shows either the plate mode ladder or the local-gap mode-density distribution, depending on geometry. Switch Material to see the Lifshitz correction for real metals vs. perfect conductors.